Icarus 290 (2017) 1–13
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Icarus
journal homepage: www.elsevier.com/locate/icarus
A Wunda-full world? carbon dioxide ice deposits on Umbriel and
other Uranian moons
Michael M. Sori∗, Jonathan Bapst, Ali M. Bramson, Shane Byrne, Margaret E. Landis
Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USA
a r t i c l e
i n f o
Article history:
Received 22 June 2016
Revised 28 January 2017
Accepted 28 February 2017
Available online 2 March 2017
a b s t r a c t
Carbon dioxide has been detected on the trailing hemispheres of several Uranian satellites, but the exact
nature and distribution of the molecules remain unknown. One such satellite, Umbriel, has a prominent
high albedo annulus-shaped feature within the 131-km-diameter impact crater Wunda. We hypothesize
that this feature is a solid deposit of CO2 ice. We combine thermal and ballistic transport modeling to
study the evolution of CO2 molecules on the surface of Umbriel, a high-obliquity (∼98°) body. Considering processes such as sublimation and Jeans escape, we find that CO2 ice migrates to low latitudes on
geologically short (100s–1000 s of years) timescales. Crater morphology and location create a local cold
trap inside Wunda, and the slopes of crater walls and a central peak explain the deposit’s annular shape.
The high albedo and thermal inertia of CO2 ice relative to regolith allows deposits 15-m-thick or greater
to be stable over the age of the solar system. We conclude that Wunda, located at low latitude (7.9° S)
and near the center of the trailing hemisphere where CO2 detections are strongest, likely contains a solid
CO2 ice deposit. We discuss prospects for similar CO2 ice deposits on crater floors on the other major
Uranian moons, and predict that they are present on Ariel, Titania, and possibly Oberon (but not Miranda
or smaller satellites). Such deposits have likely not been observed due to the limited nature of Voyager 2
image coverage.
© 2017 Elsevier Inc. All rights reserved.
1. Introduction
Voyager 2, the only spacecraft to visit Uranus, returned a wealth
of data used to study the planet, its space environment, ring system, and satellites. The imaging science system camera (wavelength coverage from 0.28–0.64 μm) was particularly useful in
studying the largest Uranian satellites, collecting images of large
regions of the southern hemispheres of the five most massive
moons: Miranda, Ariel, Umbriel, Titania, and Oberon (Smith et al.,
1986). This data set has proved essential in analyses relevant to
the moons’ geology, including but not limited to their surface ages
(Plescia, 1987a,b), orbital histories (Peale, 1988), crater morphology
(Schenk, 1989), and cryovolcanic activity (Jankowski and Squyres,
1988).
No spacecraft has visited Uranus since the 1986 Voyager 2 flyby, but new investigations of the system hold the potential for
scientific advancement for two reasons. Recent missions such as
Cassini and New Horizons have revolutionized our understanding
and intuition about the icy worlds of the outer solar system, and
Earth-based spectroscopic observation campaigns have informed
∗
Corresponding author.
E-mail address: [email protected] (M.M. Sori).
http://dx.doi.org/10.1016/j.icarus.2017.02.029
0019-1035/© 2017 Elsevier Inc. All rights reserved.
us about Uranian satellite surface compositions. Near-infrared observations made using the SpeX spectrograph (Rayner et al., 2003)
at NASA’s Infrared Telescope Facility (IRTF) revealed an asymmetry
in water ice absorption bands on the classical Uranian moons, with
deeper bands on the leading hemispheres (Grundy et al., 2003;
2006). This observation suggests less space weathered ice on those
hemispheres, removal of water ice by sputtering on the trailing
hemispheres, or burial of ice on the trailing hemispheres by another compound (Grundy et al., 2003, 2006). Such an asymmetry is
not unexpected for tidally locked satellites (like the five major Uranian moons) (e.g., Grundy et al. 1999), but a more surprising result
was the detection of carbon dioxide on the trailing hemispheres
of Ariel, Umbriel, and Titania (Grundy et al., 2003, 2006). The results form a planetocentric trend, with CO2 being most abundant
on the second innermost large satellite (Ariel) and decreasing in
abundance with distance from Uranus. Later IRTF observations confirmed these detections, and added a weak detection of CO2 on
Oberon (Cartwright et al., 2015). The innermost large satellite, Miranda, is an exception to this trend because CO2 has not been detected on this moon (Bauer et al., 2002; Gourgeot et al., 2014),
which we discuss in Section 4.3.
The planetocentric trend, along with the longitudinal preference
of CO2 at the trailing hemispheres, led Grundy et al. (2006) and
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M.M. Sori et al. / Icarus 290 (2017) 1–13
Fig. 1. Umbriel. Image 1334U2-001 taken by Voyager 2/ISS. The bright annulus inside the impact crater Wunda can be seen at the top of the picture, which is near
the equator.
Cartwright et al. (2015) to favor radiolytic production of CO2 as
the primary source of this constituent. Charged particles trapped
in magnetic field lines of Uranus will preferentially impinge on
the trailing hemispheres of prograde moons because the field
co-rotates with the planet at a faster rate than the moons’ orbital periods (Ness et al., 1986). Charged particles preferentially
strike the inner satellites because the magnetospheric density increases with proximity to Uranus (Stone et al., 1986). It has been
shown that radiolysis can produce CO2 if the target H2 O-rich material contains carbon (e.g., Gomis and Strazzulla, 2005), which
is likely the case given the low albedo of Uranian satellite surfaces (Soifer et al., 1981; Clark and Lucey, 1984). An endogenic
source of carbon is necessary to invoke radiolytic production because carbon ions have not been detected in the Uranian magnetosphere (Krimigis et al., 1986), and carbonaceous infall such as
seen on Vesta (McCord et al., 2012) is unlikely to result in the
observed longitudinal trends of CO2 (Grundy et al., 2006). Interactions between ions caught in planetary magnetic fields and icy
satellite surfaces have been documented for other bodies, e.g. Europa (Lane et al., 1981).
Umbriel (Fig. 1) is the third innermost of the five major moons
of Uranus. Crater size-frequency statistics indicate it has an ancient
(∼4 Ga) surface, with no evidence for younger regions as there is
for Miranda, Ariel, and Titania (Smith et al., 1986; Plescia, 1987a,b;
Zahnle et al., 2003). Umbriel’s surface is uniformly dark, but has
one anomalously bright feature located inside Wunda, a 131-kmdiameter impact crater (Smith et al., 1986; Helfenstein et al., 1989).
This prominent albedo outlier is an annulus, with inner diameter ∼20 km and outer diameter ∼80 km. It is tempting to speculate that Wunda is filled with CO2 ice because the feature has a
high albedo, consistent with CO2 ice, and is located on the trailing
hemisphere where CO2 was detected by previous studies (Grundy
et al., 2006; Cartwright et al., 2015). This idea is consistent with
the finding of (Grundy et al., 2006) that a relatively small fraction
of surface area could be responsible for the CO2 detections, and
with the observation that detected CO2 is segregated from other
constituents (Cartwright et al., 2015). The possibility of a Wunda
CO2 ice deposit was briefly mentioned by (Grundy et al., 2006),
but dismissed because similar features are not evident on the other
Uranian moons.
In this study, we test the hypothesis that the bright feature inside Wunda is a deposit of solid, slab-like CO2 ice. We model Umbriel’s surface temperatures as a function of latitude and surface
properties using a 1D thermal conduction model. These temperatures serve as inputs to a ballistic transport model, which we use
to study the evolution and lifetimes of CO2 particles on Umbriel.
We include the effects of the increased albedo and thermal inertia of CO2 ice relative to the rest of the surface, which lead to
greater CO2 ice stability, and the effect of Wunda’s crater morphology, which leads to the annular shape of the bright feature. We
discuss implications for the history of Umbriel and other Uranian
moons, including reasons why similar features are not observed on
Miranda, Ariel, Titania, and Oberon.
The approach outlined above follows studies of other airless
bodies (e.g., Butler, 1997; Palmer and Brown, 2008). Transport
of CO2 on airless icy satellites has been actively observed by
the Cassini spacecraft on the Saturnian satellites Rhea and Dione
(Teolis et al., 2010; Teolis and Waite, 2016). Permanently shadowed
cold traps with polar craters have been observed to contain ices on
Mercury (e.g., Neumann et al., 2013) and the Moon (Feldman et al.,
1998). However, while partial shadowing during a Uranian year
(∼84 Earth years) may contribute to surface temperature distributions, permanently shadowed regions analogous to polar craters
on Mercury and the Moon are not possible on the Uranian satellites due to the high obliquity of the Uranian system (98°). The
distribution of insolation as a function of latitude on Umbriel is
critically different compared to low-obliquity bodies, and our work
has implications for the transport of volatile distribution on highobliquity bodies in general.
2. Thermal modeling
2.1. Thermal modeling methods
Voyager 2 s infrared interferometer spectrometer (IRIS) derived
temperatures for Uranus, Miranda, and Ariel, but could not resolve Umbriel or the other Uranian moons (Hanel et al., 1986). We
thus estimate the temperatures on Umbriel by use of a thermal
model. The IRIS measurements of 86 ± 1 K for southern Miranda
and 84 ± 1 K for southern Ariel represent maximum temperatures
because they were taken at the subsolar points on each moon, and
the Voyager fly-by occurred during southern summer. We expect
Umbriel’s maximum temperature to be similar but slightly greater
due to its darker surface. We compare our maximum modeled
temperatures to the IRIS results as an approximate way to validate
our model.
We use a 1D thermal model to simulate the energy balance
at the surface from blackbody radiation and thermal conduction
with subsurface layers, solving the heat equation with the CrankNicolson method (Crank and Nicolson, 1947). Using the orbital
properties of Uranus and the rotation rate and pole-vector of Umbriel, we calculate solar fluxes at each latitude and local time on
the moon’s surface throughout a Uranian year. We assume flat terrain except for model runs that include crater geometry as discussed below. Thermal conduction is simulated by solving the thermal diffusion equation at model layer boundaries, with the top
layer in radiative equilibrium and a zero flux lower boundary condition.
Parameters relevant for our models are shown in Table 1. Thermal inertia values have not been directly estimated for Umbriel.
We select a nominal value of I = 15 J m−2 K−1 s−0.5 for Umbrielian
regolith based on observations of the thermal emission of the sur-
M.M. Sori et al. / Icarus 290 (2017) 1–13
3
Table 1
List of parameters used in our thermal and ballistic transport models. Some parameters have
values listed for icy regolith (marked with an r) and CO2 ice (marked with a C).
Symbol
Parameter
Value
Hrim
Hpeak
D
d
Dpeak
hpeak
k
I
c
E
p
i
g
s
Rim horizon height
Central peak horizon height
Rim-to-rim crater diameter
Rim-to-floor crater depth
Central peak diameter
Central peak height
Thermal conductivity
Thermal inertia
Heat capacity
Eccentricity
Vapor pressure
Sublimation rate
Umbriel surface gravity
Velocity probability distribution
Launch angle
Gravitational constant
Umbriel mean radius
Emissivity
Bond albedo
Temperature
CO2 molar mass
Umbriel mass
Semi-major axis
Travel distance
Boltzmann constant
Gas constant
Azimuth
True anomaly
Eq. 1
Eq. 2
131 km
6.7 km
20 km
4.0 km
0.0 0 017 W m−1 K−1 (r), 0.55 W m−1 K−1 (C)
15 J m−2 K−1 s−0.5 (r), 940 J m−2 K−1 s−0.5 (C)
837 J kg−1 K−1 (r), 1070 J kg−1 K−1 (C)
Eq. 7
Eq. 4
Eq. 3
0.23 m/s2
Eq. 3
Random between 0–90°
6.67e-11 m3 kg−1 s−2
584.7 km
0.95 (r), 0.95 (C)
0.1 (r), 0.5 (C)
Thermal model result
44 g/mol
1.17e21 kg
Eq. 6
Eq. 9
1.38e-23 J/K
8.31 J/K mol
Random between 0–360°
Eq. 8
θ
G
r
ε
A
T
μ
M
a
dball
kb
R
ω
faces of the icy satellites of Saturn by Cassini’s composite infrared
spectrometer (Howett et al., 2010). Surfaces of icy satellites tend
to have such low thermal inertias because of low surface pressures (i.e., airless) and high regolith porosity, which both decrease
thermal conductivity. We use a Bond albedo of A = 0.1 for Umbriel’s surface derived from Voyager image analysis (Veverka et al.,
1987; Buratti et al., 1990) and Hubble Space Telescope observations (Karkoschka, 2001a). We assume a thermal conductivity of
k = 0.0 0 017 W m−1 K−1 , a surface density of ρ = 1615 kg m−3 , a
heat capacity of c = 837 J kg−1 K−1 , and an emissivity of ε = 0.95
(based on values for Saturnian satellites (Howett et al., 2010)).
If sufficient CO2 ice collects on a local region of the surface,
the optical and thermal properties of CO2 ice are used as input
parameters. This would be the expected scenario for the prominent bright annulus in Wunda, if it were indeed CO2 ice. For these
cases, we assume A = 0.5, ε = 0.95, k = 0.55 W m−1 K−1 , ρ = 1600 kg
m−3 , and c = 1070 J kg−1 K−1 , resulting in a thermal inertia of
I = 970 J m−2 K−1 s−0.5 . This thermal inertia is a conservative choice
with regards to ice stability because it is likely that thermal conductivity of solid CO2 increases at the low temperatures found
in the Uranian system (Sumarokov et al., 2003). Amorphous CO2
ice may have different thermal properties but is not expected
at temperatures relevant on Umbriel (Escribano et al., 2013). The
albedo choice is within the allowable range of CO2 ice reflectivities (Wood et al., 1971) and the measured albedo of Wunda’s
bright deposit (Smith et al., 1986). We will argue that choosing
other allowable CO2 ice albedos does not alter our conclusions (see
Section 4.1).
Another complexity that may be important is crater morphology. Impact craters may be sites of preferential volatile accumulation on some planetary surfaces (e.g., Banks et al., 2010). Although
not well-resolved in the Voyager imagery, we expect Wunda to be
a complex crater with a central peak based on its size (Schenk,
1989). This expectation is strengthened by Cassini observations:
Dione, a moon of Saturn with comparable density and surface
gravity to Umbriel, has impact craters of similar diameters to
Wunda (e.g., the ∼140 km diameter crater Dido) that have clearly
resolved central peaks (Moore et al., 2004). The geometry of a
complex crater with a central peak has three relevant thermal effects. First, the sloping sides of the crater walls and peak experience different insolation. This effect can be a warming or cooling effect depending on the orientation of the slope. Second, the
crater walls and peak may provide shade to the adjacent crater
floor. This effect is a cooling effect because it blocks direct insolation. Third, reflected sunlight and thermal emission from the walls
and the central peak may be important. This effect is a warming
effect because it provides additional incoming radiation. Whether
these three effects cumulatively result in net warming or cooling
for different parts of Wunda requires careful thermal modeling.
The annular shape of Wunda’s bright deposit could be the result of the crater’s complex morphology, where the dark center
represents the central peak. In this case, the base of the central
peak has a diameter equal to the inner radius of the annulus,
∼20 km (Smith et al., 1986). Such a size is comparable to that
of Herschel on Saturn’s moon Mimas, a 135-km-diameter impact
crater with a depth of 10–12 km, a central peak height of 6–8 km,
and a central peak base diameter of 30 km (Moore et al., 2004).
We expect topographic relief of Wunda to be less than that of Herschel on the basis of photoclinometry measurements of Saturnian
and Uranian satellites (Schenk, 1989). To obtain a simple estimate
of Wunda’s geometry, we scale down the Herschel depth values
by 2/3, in accordance with the 20 km vs. 30 km values of central
peak base diameters. We perform thermal model simulations for
a complex crater at Wunda’s location with a diameter of 131 km,
a depth of 6.7 km, a central peak base diameter of 20 km, and a
central peak height of 4 km. We discuss the effects of uncertainties in crater dimensions in Section 4.1. These dimensions yield
wall slopes of 14.7°; the fact that other icy satellites with topographic observations have similar crater wall slopes (Bray et al.,
2008) lends confidence to our estimates.
In addition to the slopes of the crater wall and central peak receiving a different incidence angle of incoming sunlight, the topog-
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M.M. Sori et al. / Icarus 290 (2017) 1–13
Fig. 2. Horizon height as seen from Wunda’s floor as a function of viewing azimuth
for various distances along a line spanning from the crater center to the north crater
wall. The dotted and dashed lines show the distances where the fraction of sky
unobstructed by topography is at a minimum and maximum, respectively. The peak
in horizon height centered at β = 180° (i.e., facing south) represents the effect of the
crater’s central peak, while the peak centered at β = 0° (i.e., facing north) represents
the effect of the crater rim.
Fig. 3. Temperature results throughout one Uranian year for Umbriel’s surface at
the equator (blue), north pole (red), and south pole (green). Thickness of the lines
represent diurnal variations. LS = 0 corresponds to spring equinox. (For interpretation of the references to colour in this figure legend, the reader is referred to the
web version of this article.)
raphy can provide shade to parts of the crater floor and exchange
reradiated heat with the crater floor. To quantify these effects, we
must calculate the horizon height at each point of interest on the
crater floor. The horizon height due to the crater rim Hrim for an
observer at any given point on the crater floor (x, ϕ in polar coordinates where north of center is ϕ = 0 and ϕ increases clockwise),
facing a particular direction β (angle an observer on the crater
floor is facing clockwise-relative to north), is given by:
⎛
Hrim
d
= arctan⎝ ×
x
⎞
cos(π − ϕ + β ) +
1
D2
4x 2
⎠
−
sin2
(π − ϕ + β )
(1)
D is the rim-to-rim crater diameter, d is the rim-to-floor crater
depth. Similarly, the horizon height due to the central peak Hpeak
is given by:
H peak
D peak − 2xsin(|π − ϕ + β| )
h peak
= arctan
×
D peak
xcos(|π − ϕ + β| )
(2)
Dpeak is the central peak diameter, hpeak is the central peak
height (set to 0 when |π - ϕ + β | > arcsin (D/2x) to avoid negative values). Eqs 1 and 2 are derived in Appendix A. For any given
location on the crater floor (x and ϕ ) and direction (β ), the appropriate horizon height to consider is whichever of Hrim and Hpeak is
greater. Horizon heights are used to calculate if a particular location is shadowed at any given time step in the thermal model, and
if so, we set solar insolation to zero.
For a surface with topography, portions of the sky may be obstructed. Thus, not all emitted radiation escapes to space; some of
it is exchanged with surrounding topography. We integrate horizon heights over all β (0–360°) to obtain the fraction of sky that
is unobstructed to the observer (Fig. 2). For cases on sloped terrain (the crater walls or central peak), the obstructed portions of
the sky were set to radiate as a surrounding flat surface, modeled
independently. Surrounding terrain additionally reflects some solar
radiation that we add to the direct solar radiation. The shadowing
and reradiation effects described in the previous two paragraphs
are cooling and warming effects, respectively.
Fig. 4. Temperature results throughout one Uranian year at Wunda’s latitude (7.9°
S) for a surface with no topography and the thermal properties of regolith (red) and
a surface with no topography and the albedo and thermal inertia of CO2 ice (green).
LS = 0 corresponds to spring equinox. (For interpretation of the references to colour
in this figure legend, the reader is referred to the web version of this article.)
2.2. Thermal modeling results
Resulting output temperatures for the thermal properties of regolith on a sphere of Umbriel’s radius (i.e., effects of topography
are not included) are shown in Fig. 3. Note the maximum temperatures (∼90 K) are close to the IRIS results of Hanel et al.,
(1986), supporting the validity of our model. When simulating regolith temperatures, we use these thermophysical properties at all
depths. Buried layers of ice at depths of 0.5 and 5 m were found
to have no detectable effect on surface temperature results. As
discussed in Section 2.1, if Wunda’s annulus represents a deposit
of CO2 ice, it has different thermal properties and thus temperatures. Fig. 4 shows surface temperatures throughout a Uranian
year for flat terrain at Wunda’s location (7.9° S, 273.6° E) under
both sets of thermal properties (dark regolith and CO2 ice). The
M.M. Sori et al. / Icarus 290 (2017) 1–13
5
Fig. 5. Sensitivity of thermal model results to thermal inertia. Maximum (solid
lines) and mean (dashed lines) temperatures throughout a Uranian year for flat topography at Wunda’s latitude (7.9° S) for cases of regolith (red) and CO2 ice (green).
The break in x-axis is for clarity, as thermal inertia values in the missing intermediate domain are unlikely to represent regolith or ice. (For interpretation of the
references to colour in this figure legend, the reader is referred to the web version
of this article.)
presence of a CO2 ice deposit suppresses local maximum temperatures by ∼30 K. This finding is similar to the effect that ices have
on local temperatures on Pluto (Earle et al., 2016), but the temperature difference is greater in our model because we took into
account both the increased albedo and thermal inertia of ice deposits, whereas (Earle et al., 2016) focused on studying the effects
of variable albedo.
We discuss the effects of uncertainties in model parameters on
our conclusions in Section 4.1, but thermal inertias are especially
uncertain due to the lack of observationally derived values for any
Uranian moon and deserve quantitative consideration. We perform
sensitivity tests on the values we list above (and in Table 1) by
running thermal models for a range of thermal inertias. Best-fit
thermal inertia values derived by (Howett et al., 2010) for large
Saturnian satellites range from 8 J m−2 K−1 s−0.5 (for Rhea’s trailing hemisphere) to 20 J m−2 K−1 s−0.5 (for Iapetus’ trailing hemisphere), so we test values within this range around our nominal
value of 15 J m−2 K−1 s−0.5 . We similarly test a range of values
for the thermal inertia of CO2 ice, from 30 0–110 0 J m−2 K−1 s−0.5
around our nominal value of 970 J m−2 K−1 s−0.5 . Thermal inertia is a function of thermal conductivity, material density, and heat
capacity; our test effectively measures the sensitivity of results to
conductivity because it is generally the least constrained of these
three parameters. Results of our sensitivity test are shown in Fig. 5.
Because sublimation rate increases rapidly with temperature, we
plot the maximum annual temperature in addition to the annualaverage temperature for each case. Maximum temperature only
varies between 86.6–82.2 K as the thermal inertia varies between
8–20 J m−2 K−1 s−0.5 for regolith and between 57.8–54.6 K as thermal inertia varies between 30 0–110 0 J m−2 K−1 s−0.5 for CO2 ice.
We use surface temperature results to predict sublimation rates
of hypothetical surface ice. Sublimation rate i is a function of
the vapor pressure p, temperature T, Boltzmann constant kb , and
molecular mass μ of CO2 :
μ
i=p
2π kb T
(3)
The vapor pressure of CO2 at temperature T is given by the
Clausius–Clapeyron relation:
p = 101.3e23.102−
3148.0
T
(4)
Although experimental data measuring the vapor pressure of
CO2 are only available down to temperatures of ∼150 K, theoretical work suggests it is reasonable to extrapolate empirical relations to the lower temperatures found on Umbriel (Azreg-Aïnou,
Fig. 6. Total sublimation of CO2 ice per m2 in a Uranian year as a function of latitude on a surface with the albedo and thermal properties of icy satellite regolith.
Total sublimation for cases with the albedo and thermal properties of CO2 ice are
7–9 orders of magnitude less and do not appear in the Fig..
2005). Using Eqs. 3 and 4, we calculate the sublimation rate of CO2
ice throughout a Uranian year at every 5° of latitude. We integrate
these sublimation rates through time to estimate the total amount
of CO2 sublimated on Umbriel’s surface in one Uranian year.
Annual sublimation results, for a perfect sphere with no topography and the thermal properties of regolith, are shown in Fig. 6.
Such a set-up, also used in some of our ballistic transport models
(Section 3), describes an Umbrielian surface where small grains of
CO2 ice are everywhere mixed into the regolith. This is likely not a
realistic scenario for present-day Umbriel, and the absolute values
of our model results describing the CO2 budget as a function of
location may not be correct. Instead, we argue that our model results use these sublimation rates to accurately describe the relative
sources and sinks of CO2 across Umbriel’s surface. We additionally
calculate sublimation rates for solid, slab-like CO2 ice (A = 0.5 and
I = 940 J m−2 K−1 s−0.5 ) at Wunda’s location and find them to be 7–
9 orders of magnitude less than sublimation rates for the regolith
case. This large decrease in sublimation is due to a much lower
maximum temperature: ∼55 K for CO2 ice compared to ∼85 K for
regolith at this location (Fig. 4).
Our thermal inertia sensitivity test (Fig. 5) indicates that
at Wunda’s location, total sublimation in one Uranian year
varies between 24.8–3.3 kg/m2 as thermal inertia varies between
8–20 J m−2 K−1 s−0.5 for regolith and between 1.5 × 10−6 –
2.1 × 10−7 kg/m2 as thermal inertia varies between 30 0–110 0 J m−2
K−1 s−0.5 for CO2 ice.
3. Ballistic transport
3.1. Ballistic transport methods
We use a ballistic transport model to understand the fate and
evolution of sublimated CO2 molecules, similar to an approach
used to study CO2 on Iapetus (Palmer and Brown, 2008). We assume that when a molecule sublimates at a particular time step,
it takes a single suborbital hop on a ballistic trajectory and that
sublimated molecules do not collide. We neglect effects of solar radiation pressure on particles. The destination of a sublimated molecule is therefore determined by its initial location, velocity, and direction. These assumptions are similar to those of
Palmer and Brown (2008).
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M.M. Sori et al. / Icarus 290 (2017) 1–13
Fig. 7. Velocity distributions of sublimated CO2 molecules on Umbriel at three different temperatures. 20 K and 90 K are the approximate minimum and maximum
temperatures we calculate on Umbriel. Vertical dashed line represents Umbriel’s escape velocity.
We assume the distribution of initial directions is isotropic. That
is, the launch angle θ at which a molecule leaves the surface relative to horizontal is randomly chosen from values between 0°–90°
(weighted towards low angles with a cos θ term to keep the distribution isotropic), and the azimuth is randomly chosen from
values between 0°–360° (uniformly). The initial velocity v is chosen from a half-Maxwell-Boltzmann probability distribution s and
is a function of the temperature T, molecular mass μ, and Boltzmann constant kb :
s = 4π
2
μ 1.5 2 −2μv
v e kb T
2π kb T
(5)
We show these velocity distributions for a few relevant temperatures on Umbriel in Fig. 7. The launch angle θ and velocity distribution s can be modified to include the effects of surface roughness, but this effect has been shown to be minimal and so we
choose θ and s as described above (Butler, 1997). The distance dball
a sublimated molecule travels is found using orbital parameters of
the molecule’s path: the semi-major axis a, the eccentricity E, and
the true anomaly ω. These parameters are a function of the gravitational constant G, the mass of Umbriel M, the radius of Umbriel
r, and the initial direction and velocity:
a=
2
r
1
v2
− GM
E=
1−
(6)
r (2a − r )sin2 φ
a2
(7)
a − aE 2 − r
ω = arccos
(8)
dball = (2π − 2ω )r
(9)
rE
We calculate the landing point of each sublimated CO2
molecule using these equations.
We consider the loss of CO2 molecules from Umbriel
via
Jeans escape. The escape velocity of Umbriel, given by
2gr, is
517 m/s. If a CO2 molecule in our simulation is assigned a velocity
v > 517 m/s from the velocity distribution s, we remove it from the
system and assume it is permanently lost. In Fig. 8, we show the
fraction of sublimated molecules that escape via this mechanism
as a function of temperature.
Fig. 8. Fraction of sublimated CO2 molecules that escape Umbriel via Jeans escape
as a function of temperature. Note the logarithmic y-axis.
3.2. Ballistic transport results
We combine the temperature results from our thermal model
(Section 2.2) with our ballistic transport method to simulate the
evolution of CO2 ice over Umbriel’s surface. In our most basic
model, we assume Umbriel’s surface has the thermal properties
of regolith and no relevant thermal effects from topography. We
take a Monte Carlo approach to determine the distribution of landing locations of sublimated molecules for a particular starting location and temperature. We simulate 105 sublimated molecules at
every 5° of latitude for the temperatures that latitude experiences
at some point throughout a Uranian year, according to our thermal
results. Using Eqs 5–9, we tracked the latitude where each of these
105 molecules landed. We then walked through time steps of 50 0 0
seconds at every 5° of latitude and sublimated the appropriate
amount of CO2 at each latitude and time step according to Eqs 3–4,
using our distribution of the destinations of sublimated molecules
to determine the ending locations of sublimated CO2 . This time
step is sufficient to resolve diurnal temperature variations (an Umbrielian day is ∼350,0 0 0 s), but coarser than the average flight time
of sublimated molecules (∼10 0 0 s) so that molecules are unlikely
to require several time steps to complete their trajectories. We kept
track of the net change in CO2 at every 5° of latitude throughout this process and integrated the results throughout one Uranian
year to calculate the annual CO2 budget for the entire moon. These
results are shown in Fig. 9a.
Our results show that CO2 ice mixed with dark Umbriel regolith
is rapidly depleted from the polar regions. Latitudes greater than
50° experience a net loss of CO2 and latitudes less than 50° experience a net gain of CO2 . Regions at a latitude of 35° experience
the largest gain of CO2 but this is a result of our initial condition
of uniform CO2 . Low latitudes experience the least amount of annual sublimation, but are farthest from the polar regions that are
the biggest sources of CO2 .
We performed the same simulation, but with an initial condition where CO2 was already depleted from latitudes greater than
50° Results from this simulation reveal that the latitudes that experience a net gain of CO2 narrow over time as expected (Fig. 9b).
Further simulations of this type show that as the initial source region is narrowed in latitude, the preferred latitude for CO2 is the
equator. CO2 is rapidly mobilized toward very low latitudes on Umbriel over geologically short timescales (kyrs). Simulations with an
M.M. Sori et al. / Icarus 290 (2017) 1–13
7
Fig. 9. Net CO2 ice flux per m2 in a Uranian year across Umbriel with no topography (the effects of complex crater topography are shown in Fig. 10). Top row depicts the
fluxes on Umbriel’s surface; the south and north poles are labeled with an S and N respectively. Bottom row displays the same results, but simply as a function of latitude.
(a) is for an initial condition of CO2 ice everywhere on a surface with the properties of Umbriel regolith. (b) is the same as (a), but the initial condition only has CO2 ice
equatorward of 50°, i.e. after the polar regions have been depleted. Note the different scales in (a) vs. (b) and the narrower region of positive flux; CO2 migrates towards the
equator over time, and polar regions are not sinks for CO2 .
initial condition of CO2 ice only mantling the trailing (where it
would be produced by radiolysis) or leading hemisphere have the
same behavior as shown in Fig. 9, but with the resulting CO2 distribution concentrated in the initial hemisphere.
We consider an additional case similar to the ones above, but
with a complex crater at Wunda’s location with the morphological parameters described in Section 2. We use the same technique
described above. Note there is no longer longitudinal symmetry
in this model, and in contrast to the results shown in Fig. 9, results may vary for locations at the same latitude. These results are
shown in Fig. 10. CO2 fluxes are calculated at 40 points throughout
the crater: for each of 8 directions (N, NE, E, SE, S, SW, W, and NW)
we calculate the flux for a point midway up the crater wall, a point
midway up the central peak, a point on the crater floor at the base
of the central peak, a point on the crater floor at the base of the
crater wall, and a point halfway between those two floor points.
In Fig. 10, the flux results on the crater wall and central peak are
interpolated in a nearest-neighbor sense, and results on the crater
floor are interpolated with triangulation-based linear interpolation.
Because our initial condition of CO2 distributed everywhere on
Umbriel in large quantities is an idealized case, it is important
to interpret the fluxes in Fig. 10 in a relative sense. The floor of
Wunda is a relative sink of CO2 compared to the flat terrain surrounding it (Fig. 10b); the cooling effect of shadowing slightly exceeds the warming effect of reradiation from surrounding topography. The crater walls and central peak are relative sources of
CO2 compared to flat terrain (both on the crater floor and otherwise). Maximum temperatures on Wunda’s walls and central peak
are only ∼1.5 K warmer than maximum temperatures on the crater
floor, but integrated over a Uranian year, the slight temperature
difference can lead to a factor of ∼2 difference in sublimation (for
the most extreme case: the poleward facing side of the central
peak vs. the coldest part of the crater floor). The crater floor experiences ∼8% less sublimation than flat terrain just outside the
crater. These effects could provide an explanation for the annular
shape of Wunda’s bright deposit.
Wunda may be a preferred site of CO2 deposition compared to
other low-latitude complex craters due to its proximity to the center of the trailing hemisphere, where CO2 may be most efficiently
produced by radioalysis. Wunda’s bright deposit contains a lowalbedo streak in its southern half (Fig. 1). Although our results reveal that the northern crater floor is slightly favored for CO2 deposition compared to the southern crater floor, this effect is very
minor, and it is more likely that topography not accounted for in
our idealized simulations causes the low albedo streak.
4. Discussion
4.1. Evolution of wunda’s bright deposit
We have shown that CO2 ice migrates quickly toward the equator on Umbriel, but total sublimation per Uranian year is still high
8
M.M. Sori et al. / Icarus 290 (2017) 1–13
sis (Grundy et al., 2006; Cartwright et al., 2015), it may still be produced today. We ran the simulation described above for CO2 particles starting at a random location on the trailing hemisphere, and
found similar percentages of ∼10% and ∼90% for CO2 molecules
that deposit in Wunda and escape the system respectively. Thus,
if Umbriel has a production rate > 5 × 10−6 kg/s (or in terms of per
unit area, 10−18 kg m−2 s−1 ), the Wunda deposit should be growing
today.
4.2. Model uncertainties
Fig. 10. (a) Topographic profile (vertically exaggerated × 3) through the center of
our model’s complex crater, such as the dashed line in (b). (b) Results showing the
total CO2 flux per m2 in a complex crater of Wunda’s size and location, for an initial
condition of CO2 everywhere on Umbriel’s surface with the properties of regolith.
We set the minimum flux on the color scale to be 5 kg/m2 for clarity, but the southfacing slope of the central peak experiences flux as low as −2.4 kg/m2 . The crater
floor is a CO2 trap relative to the crater walls and central peak.
(∼6 kg/m2 or greater), even at the global minimum at the equator
(Fig. 6). This is a consequence of the high obliquity of the Uranian system. However, the sublimation rate at low latitudes for
a surface with the albedo and thermal inertia of CO2 ice is very
low, ∼3 × 10−7 kg/m2 each Uranian year (∼84 terrestrial years). To
understand whether this is sufficiently low to allow survival of a
CO2 deposit over long geologic time scales, we need to understand
the CO2 production rate and what happens to the molecules sublimated from the deposit.
To quantify these effects, we ran a simulation with an initial condition similar to an idealized present-day Umbriel. In this
model, Umbriel is a sphere with no topography and the thermal
properties of dark regolith everywhere, except at Wunda’s location, where there is a complex crater with a bright annulus of CO2
ice albedo and thermal inertia. We determined the fate of a sublimated CO2 molecule under this condition. The molecule ballistically hops over time until it eventually returns to Wunda’s CO2
ice deposit or escapes Umbriel completely (because one of its hops
exceeds escape velocity). We ran a Monte Carlo simulation of 105
sublimated molecules (we found that increasing the number of
molecules does not significantly alter the following results). We
find that ∼15% of sublimated molecules eventually find their way
back to Wunda, while ∼85% escape the system via Jeans escape.
Multiplying this percentage by the sublimation rate of the bright
deposit and its area, we find that the entire area of Wunda loses a
total of ∼1200 kg of CO2 ice every Uranian year (a little less than a
m3 ), for an average of 5 × 10−7 kg/s, or, equivalently, ∼1.6 × 10−10 m
of ablation every Uranian year. We estimate the minimum thickness of a CO2 ice deposit that justifies our use of CO2 thermal
properties to be the annual skin depth, ∼15 m. At this minimum
thickness, the entire bright deposit has a mass of 2 × 1014 kg and
therefore would persist over the age of the solar system even if it
experiences no CO2 replenishment.
This result has implications for the future evolution of Wunda’s
bright feature. If the source of detected CO2 on Umbriel is radioly-
Limited spacecraft data on Umbriel leads to uncertainty in our
input parameters (Table 1). Our main qualitative result—that CO2
ice is readily mobilized on Umbriel, migrating quickly to low latitudes, and stable over Gyr timescales if it accumulates in a single patch—is robust even though varying some of these parameters may change the resulting sublimation distribution. For example, we use an albedo of A = 0.5 for CO2 ice, a value consistent
with that observed for the bright feature in Wunda (Smith et al.,
1986). However, reasonable albedos of CO2 ice could range from
0.5–0.7 (Wood et al., 1971; Grundy et al., 2006), but the effect of
an increased albedo would only increase the stability of CO2 ice
deposits further.
Our nominal thermal inertia value (970 J m−2 K−1 s−0. 5) for CO2
assumes that the CO2 deposit is in the form of solid, cohesive ice.
Although we have shown sublimation is a slow process on Umbriel, grain growth is as well (Clark et al., 1983), leading to questions as to how long it would take radiolytically-produced CO2 to
form non-porous ice. However, thermal model results for the CO2
deposit have a relatively low sensitivity to the exact thermal inertia value (Fig. 5). The maximum temperature of a CO2 deposit
during a Uranian year changes by only ∼3 K as a result of varying
thermal inertia within a large range of 30 0–110 0 J m−2 K−1 s−0.5
(the effect that this range has on sublimation rates is discussed in
Section 2). We find that the key needed for stability of a CO2 deposit over geologic time in Wunda is for the thermal inertia to be
greater than that of Umbriel’s regolith. The thermal inertia does
not necessarily need to be so high that it represents a completely
non-porous, cohesive slab of CO2 ice.
There are a few effects not considered in our models. The one
with the most potential importance is ultraviolet (UV) photolysis. UV radiation has to ability to dissociate CO2 on airless body
surfaces (e.g., Gerakines and Moore, 2001) into carbon monoxide
and oxygen. While this effect has been estimated to be relatively
rapid on the surface of Uranian satellites (Grundy et al., 2006), it
remains unclear if UV dissociation is a dominant process in this
context because it should not produce the observed planetocentric (more CO2 on inner satellites) or longitudinal (more CO2 on
trailing hemispheres) trends (Cartwright et al., 2015). A possible
counteracting factor is UV-induced CO2 synthesis, in which absorption of photons by H2 O ice mixed with carbon-rich material
produces CO2 (e.g., Charakov et al., 2001). Laboratory experiments
have shown that UV-irradiated cometary ice analogs can produce
CO2 and other compounds (Bernstein et al., 1995). Further study
on this issue is warranted because such experiments were done at
temperatures colder than the Uranian moons and because the penetration depths involved in UV-induced synthesis is different than
that involved in radiolytic production (Delitsky and Lane, 1998). If
the rate of UV dissociation is dominant over that of radiolytic production and/or UV synthesis, another source of CO2 will need to
be invoked.
Three neglected effects in our models are adsorption, Uranusshine, and the possibility of changing orbital parameters. While
adsorption of CO2 molecules to regolith grains likely occurs to
some degree, we model sublimation as the dominant process (e.g.,
Palmer and Brown, 2008). Adsorption will not affect the ultimate
M.M. Sori et al. / Icarus 290 (2017) 1–13
fate of CO2 molecules, only the time it takes an individual molecule
to reach that fate (i.e., Jeans escape or caught in a low-latitude cold
trap).
Uranus-shine is a minor effect because of Uranus’ low energy
balance (Pearl et al., 1990), which is perhaps due to the presence of
thermal boundary layers in the planet’s interior (Nettelmann et al.,
2016). Approximating Uranus as a Lambert sphere, we integrate
flux reflected toward Umbriel’s sub-Uranian point and calculate it
to be only 0.2% of the solar flux, a number that is a factor of ∼20
too large for Wunda since it lies far from the center of the Uranusfacing hemisphere. Our models use the current orbital parameters
and obliquity for Uranus and Umbriel, which we assume are representative of their long-term values. If these parameters change
significantly over solar system history, the preferred locations of
CO2 ice deposition may change accordingly.
Our thermal models reveal that shadowing from crater walls
and the central peak cools down the crater floor relative to other
flat terrain. This effect is small: for the example in Fig. 10, Wunda’s
floor has a net CO2 flux 8% greater than uncratered terrain at the
same latitude. However, there is large uncertainty in crater dimensions, as we extrapolated depth and central peak height of a similar crater on Mimas. Because we extrapolate from the lower values given by Moore et al. (2004), our model crater depth (6.7 km)
and central peak height (4 km) are likely underestimates. Our results on the effect that crater geometry has on CO2 ice deposition
are therefore conservative. The cooling effect on the crater floor
should be stronger than our models predict (due to higher topography providing more shade), as should the warming effect on the
walls and central peak. Increased thermal inertia at Wunda’s location relative to other parts of Umbriel could also favor CO2 deposition because it would suppress maximum temperatures and thus
inhibit sublimation.
Grundy et al. (2003, 2006) & Cartwright et al. (2015) generated
synthetic spectra in an attempt to estimate the quantity and nature
of CO2 needed to produce the telescopic observations. For Umbriel,
the best fit mixture of Cartwright et al. (2015) involves ∼8% of the
surface area of the trailing hemisphere covered with a thin CO2
layer. Wunda’s bright deposit only represents ∼1% of the surface
area of the trailing hemisphere. It is possible that there are thus
multiple solid CO2 ice deposits similar to Wunda’s contributing to
the CO2 detections. Indeed, our model as described does not favor
Wunda over other equatorial craters with the same size and morphology in regards to CO2 ice stability. We do not favor a scenario
of multiple prominent solid CO2 ice deposits within craters for two
reasons. Firstly, other observed relatively bright areas in craters
on Umbriel are far lower in albedo compared to Wunda’s deposit
(Helfenstein et al., 1989), and can be explained in a simpler way
by excavation of less space-weathered material. Secondly, needing
to invoke multiple bright deposits similar to Wunda’s makes Voyager 2 s observations of a single bright feature on a single Uranian
satellite statistically unlikely (see Section 4.3).
Instead, we favor the case of Wunda being the only bright,
solid, locally concentrated CO2 ice deposit on Umbriel. It is possible that Wunda is preferred for deposition over other low-latitude
craters because it is the one with the most favorable morphology,
or because CO2 is produced most strongly at the center of the trailing hemisphere (near Wunda’s location). Although the ∼8% surface area of Cartwright et al. (2015) is the best fit model based
on spectra alone, it is not the only allowable model. Given the uncertainties involved, we argue that Wunda’s deposit best fits the
combination of the spectral observations and Voyager 2 image observations. We note that our favored scenario does not preclude
radiolytic production; CO2 mixed in with regolith grains outside of
Wunda on the trailing hemisphere may still contribute to spectral
detections.
9
4.3. Other uranian moons
CO2 is detected on Ariel, Umbriel, Titania, and perhaps Oberon
(Grundy et al., 20 03, 20 06, Cartwright et al., 2015), but only Umbriel has a feature as bright as Wunda’s. If Wunda contains a
CO2 ice deposit, we require an explanation for why similar bright
features are not seen on the other four major Uranian moons, a
problem pointed out by Grundy et al., (2006). Smooth deposits
on crater floors have been observed on Oberon (Helfenstein et al.,
1991), but they are dark, both in an absolute sense and relative to
the surrounding terrain, with albedos as low as ∼0.05 (Smith et al.,
1986). Broadly speaking, the two possibilities are: solid, bright, locally concentrated CO2 ice deposits are not currently present on
the surface of these other satellites, making Umbriel’s feature a
special case; or that such deposits are present on the other satellites but not yet observed, or obscured by sublimation lag.
There are plausible reasons to believe that CO2 deposits do not
exist on all of the other four large satellites despite the fact that
imagery only exists for less than half their surfaces. The observations of (Grundy et al., 2006 & Cartwright et al., 2015) did not
target Miranda, but if the radiolysis hypothesis is correct, CO2 production should be greatest there as it is the innermost of the five
large moons. However, we do not expect segregated CO2 ice deposits on Miranda because of Jeans escape. Miranda’s escape velocity of 193 m/s is far less than that of the other four moons. Comparing that value to expected velocity distributions of sublimated
molecules on Uranian satellites (Fig. 7), it is clear Miranda would
rapidly lose CO2 to space, with ∼50% of sublimated molecules escaping the system before even a single suborbital ballistic hop. In
order to resist sublimation and atmospheric escape, any CO2 on
Miranda would need to be trapped as e.g. clathrates, as has been
observed on other low mass moons (e.g., Clark et al., 2005). Similarly, all Uranian moons outside the classical five are also far too
small to retain segregated CO2 . The next largest satellite, Puck,
has an escape velocity of ∼67 m/s (derived from measurements of
Karkoschka (2001b)). This argument does not apply to Ariel, Titania, or Oberon, which have higher escape velocities than Umbriel.
CO2 ice deposits are uncertain on Oberon. There is debate as to
the presence of CO2 detection on Oberon: (Grundy et al., 2006)
did not report a successful detection from Oberon observations,
while Cartwright et al. (2015) do argue for a statistically significant
(though weak) detection. The weak detection may be the result of
Oberon’s distance from Uranus. Oberon spends part of its orbit outside the Uranian magnetic field, meaning that radiolytic production
is weak because the only source of ions is the solar wind during
that time. A similar argument has been made to show the minimal
effect of radiolysis in producing CO2 on Iapetus, which is outside
Saturn’s magnetic field entirely (Palmer and Brown, 2011). Thus, if
CO2 ice is present on Oberon, it may only be in a small quantity
that is unable to accumulate in a single deposit and build up a
higher albedo and thermal inertia relative to surrounding terrain.
Ariel has been observed to have at least a partially young surface with some degree of geologically-recent resurfacing (Smith
et al., 1986; Plescia, 1987b; Zahnle et al., 2003), perhaps driven
by tidal heating (Peale, 1988). It is possible that an ancient CO2
ice deposit could be resurfaced by and buried under such extrusions. However, we do not find this explanation attractive since
CO2 ice is unambiguously identified on the present-day surface of
Ariel (Grundy et al., 2003; Cartwright et al., 2015). Similarly, we
have no explanation for why such a deposit would not exist on
Titania.
Instead, we argue that CO2 ice deposits likely exist on Ariel, Titania, and possibly Oberon, but have not yet been observed. Image
coverage only spans a portion of the southern hemisphere on each
of the five moons. Pictorial maps constructed by the U.S. Geological
Survey (USGS) reveal that Umbriel is the only Uranian moon with
10
M.M. Sori et al. / Icarus 290 (2017) 1–13
Fig. 11. USGS maps showing image coverage of the five major Uranian moons. Each large circle represents the southern hemisphere of the moon, and the labels show the
longitude east; no moon has any significant coverage in the northern hemisphere. Outlined areas with an image number written inside represent areas of the moon that
are covered. Small green circle represents the location of Wunda on Umbriel. Note that Umbriel is the only moon with coverage at the equator at the center of the trailing
hemisphere (0° N, 270° E). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
image coverage on the center of the trailing hemisphere (Fig. 11),
which is the approximate location of Wunda. The USGS maps show
that none of the five major satellites have significant coverage on
their northern hemispheres. The latitude of Wunda is 7.9° S. If we
assume that each of Ariel, Titania, Umbriel, and possibly Oberon
has a bright CO2 ice deposit, and that they are equally likely to be
located at very low northern latitude as they are very low southern latitude, it is not surprising that we only have currently observed one of the three or four. If our hypothesis that Wunda contains a CO2 ice deposit is correct, we predict full image coverage
of all Uranian moons would reveal that Titania, Ariel, and possibly Oberon to have a similar low-latitude bright deposit, but not
Miranda. We argue that ice is likely to be largely concentrated in
one deposit, as observed on high-obliquity Pluto (Stern et al., 2015;
Grundy et al., 2016) and suggested by the longitudinal preference
of CO2 on Uranian moons (Grundy et al., 2006; Cartwright et al.,
2015), but further observations are needed to determine if multiple deposits of similar size could be plausible.
The spectroscopic analysis of Ariel (Grundy et al., 2003, 2006;
Cartwright et al., 2015) indicate a larger abundance of CO2 there.
Therefore, if our reasoning above is correct, the bright CO2 ice deposit we hypothesize for Ariel likely must be larger than a feature
that can fit inside a complex crater of Wunda’s size. One possibility is that ice is simply concentrated into an impact basin that
is significantly larger than Wunda. Alternatively, Ariel’s highly tectonized terrain is dominated by long canyons (Smith et al., 1986),
and may play a role. One or more canyons mantled by CO2 deposits may explain the wider range of longitudes that are suggested to contain CO2 by the spectroscopic detections (Grundy
et al., 2003, 2006; Cartwright et al., 2015).
4.4. Alternatives to ballistically transported CO2 ice
Alternative explanations for Wunda’s bright feature are less attractive than our CO2 ice deposit hypothesis. Impact crater ejecta
can be bright relative to Umbriel’s dark surface (Helfenstein et al.,
1989), but should not reside within the crater itself. Cryovolcanism
has been suggested as a possible source of CO2 on Uranian satellites in addition to radiolytic production (Cartwright et al., 2015),
and Wunda has specifically been mentioned as possibly housing a cryovolcanic deposit (Plescia, 1987a). However, early endogenic resurfacing on Umbriel is expected to involve dark material
(albedo ∼0.1) (Helfenstein et al., 1989), and later endogenic activity
is not expected (Peale, 1988). The expectation of dark extrusions
is strengthened by observations of smooth deposits in craters on
Oberon that are interpreted as extrusive (Helfenstein et al., 1991),
but are darker than the surrounding terrain and have low albedo
(as low as ∼0.05).
Another argument against extrusive CO2 in Wunda concerns the
uniformly dark nature of Umbriel’s surface. It has been suggested
that this unusual surface is the result of a blanketing of low-albedo
material (Smith et al., 1986). If this hypothesis is correct, the superposed bright material in Wunda would need to be younger,
but late-stage endogenic activity is unlikely on Umbriel based on
analysis of crater distributions (Plescia, 1987a) and orbital histories
(Peale, 1988). Extrusive CO2 would be more plausible if instead the
homogenous dark surface was representative of Umbriel’s upper
crust, which would allow endogenic activity superposed on that
dark surface to be ancient.
Whatever the nature of Umbriel’s dark material, it is possible
brighter material may be excavated from depth via impacts. Of par-
M.M. Sori et al. / Icarus 290 (2017) 1–13
ticular interest here is that CO2 has been suggested to be excavated
by impacts on Callisto (Hibbitts et al., 2002). Hibbitts et al., (2002)
interpreted the detected CO2 on Callisto as being sourced from CO2
clathrates because segregated CO2 is predicted to be thermally unstable on surfaces in the Jupiter system, but CO2 ice could be thermally stable in the Uranian system (Lebofsky, 1975). Because Umbriel has craters larger than Wunda (Malingee and Wokolo) that do
not contain similarly bright deposits, and because excavated CO2
should be associated with crater ejecta as well as within the crater
itself (as seen on Callisto), we argue that the bright material inside
Wunda is unlikely to represent a direct excavation of CO2 . However, large impacts on Umbriel could excavate CO2 that later migrates to Wunda; this process provides an additional source of CO2
to complement radiolytic production at the present-day surface.
Ices other than CO2 are worth considering as constituents of
Wunda’s bright deposit. Nitrogen ice deposits have been observed
on Pluto (Stern et al., 2015) and may exist on the surface of Neptune’s moon Triton (e.g., Hansen and Paige, 1992). Although N2 ice
has bright albedo similar to CO2 ice, it sublimates at too low temperatures to be stable on Uranus’ warmer moons throughout even
one Uranian year. Jeans escape is also more important for N2 ice
compared to CO2 ice because of its lower molecular mass. We estimate, based on Eq. 5 and our thermal model results, that ∼2% of
sublimated N2 molecules will immediately escape Umbriel, over an
order of magnitude greater than that of CO2 (< 0.1%). This escape
rate is too high for N2 to accumulate on Umbriel’s surface. Even
if a single deposit of N2 ice the size of Wunda’s bright patch did
form, we find (using the same exercise described in Section 4.1 for
CO2 molecules) that > 98% of N2 molecules sublimated from such
an ice patch escape Umbriel.
Carbon monoxide and methane ices have also been detected
on Pluto, sometimes appearing in crater floors (Stern et al., 2015;
Grundy et al., 2016). CO has a similarly high escape percentage
of sublimated molecules as N2 , and methane ice or ammonia ice
would have a much greater one (> 10%, due to their lower molecular masses). Ultimately, the mass of Umbriel (∼1.2 × 1021 kg)
is potentially high enough to retain a CO2 ice deposit over Gyr
timescales, but not a deposit composed of the other ices found in
the outer solar system.
H2 O ice has a low molecular mass compared to CO2 , but is
not volatile at temperatures relevant for Uranus, so the above argument concerning Jeans escape does not apply. Water ice composes a majority of Umbriel’s mass, so it is worth considering
the possibility that Wunda exposes a bright Umbrielian crust beneath its dark surface. However, larger, and presumably deeper
(Schenk, 1989), craters do not expose similar bright spots. Additionally, while more study is needed to precisely quantify the effects of space weathering on the Uranian satellites, it seems clear
that charged particle irradiation would darken nearly-pure H2 O deposits on Umbriel on geologically fast timescales (Thompson et al.,
1987). In contrast, a CO2 ice deposit could maintain a bright
appearance via the addition of new radiolytically produced CO2
molecules that superpose older CO2 ice that has been darkened by
space weathering effects.
5. Conclusions
We conclude based on our thermal and ballistic transport modeling, combined with Voyager 2 observations, that the bright feature inside Wunda is likely a deposit of CO2 ice. There are several
lines of evidence supporting this idea. (1) CO2 ice has been detected on the trailing hemisphere of Umbriel (Grundy et al., 2006;
Cartwright et al., 2015), and Wunda lies very near the center of the
trailing hemisphere. (2) Our thermal and ballistic transport models show CO2 ice migrates to low latitudes over geologically short
timescales, and Wunda lies nearly at the equator. (3) Umbriel occu-
11
pies a “sweet spot” in regards to CO2 volatility: it is warm enough
for CO2 on regolith to be mobilized via sublimation for migration
to cold traps, but cold enough such that a pure sequestered CO2
surface deposit is largely resistant to sublimation, and cold and
massive enough to prevent most sublimated CO2 from escaping
via Jeans escape. Our thermal models show that a CO2 ice deposit,
once accumulated sufficiently to alter its local region’s albedo and
thermal inertia from that of Umbriel’s dark regolith to that of CO2
ice, can survive over the age of the solar system. (4) The observed
albedo of the bright feature in Wunda Smith et al., 1986) is consistent with the albedo of CO2 ice (Wood et al., 1971). ((5) Wunda is
likely a complex crater with a central peak, which provides a plausible explanation for the annular shape of the bright floor deposit
due to surface slope and shadowing effects.
Lines of evidence (1), (2), and (3) also apply to Ariel, Titania,
and Oberon, although Miranda is likely insufficiently massive to
retain CO2 ice over geologically long timescales. Therefore, our results and analyses suggest that similar CO2 -filled complex craters
could be present on Ariel, Titania, and Oberon. Such craters may
simply be undiscovered because of the sparse imaging coverage of
these moons by Voyager 2 during its flyby. We note that Umbriel is
the only moon of these five major satellites where coverage exists
at the center of the trailing hemisphere, where Wunda is located.
Our results and analyses support the hypothesis that the bright annulus in Wunda is composed of CO2 ice. Subsequent high resolution telescopic observations and/or another spacecraft mission to
the Uranian system (Arridge et al., 2014) are needed to further investigate Umbriel’s anomalously bright deposit.
Acknowledgements
We thank the Voyager 2 imaging science system team for collecting data on the Uranian system that is still bearing fruit 30
years after the fly-by, and Drs. Eriksson and McGasket for their
inspiration throughout this project. We thank Richard Cartwright
and Eric E. Palmer for their comments which greatly improved the
quality of this paper.
Appendix A
This Appendix provides derivations for calculating the horizon
heights of the rim (Appendix A.1) and central peak (Appendix A.2)
of a complex crater relative to an observer standing on the crater
floor. The final equations are Eqs. 1 and 2 in the main text. Figs. A1
and A2 provide visual aid for the calculations of horizon heights
for the crater rim and central peak, respectively.
A.1. Horizon height: crater rim
Assume an observer is standing on the floor of a complex crater
of rim-to-rim diameter D and rim-to-floor depth d at a distance x
from the center of the crater (Fig. A1). The observer is located at an
angle ϕ , measured clockwise from due north, and is facing direction β measured in the same way. The lateral distance y between
the observer and the crater rim is related to the observer’s position
by the cosine rule:
D 2
2
= x2 + y2 − 2xycos(π − ϕ + β )
(A1)
Rearranged as a function of y, Eq. A1 becomes:
y2 − [2xcos(π − ϕ + β )]y + x2 −
D 2 2
=0
(A2)
12
M.M. Sori et al. / Icarus 290 (2017) 1–13
We can now use the quadratic equation to obtain solutions for
y:
y = xcos(π − ϕ + β ) ±
x2 cos2 (π − ϕ + β ) − x2 +
D 2
2
(A3)
Note that because the crater radius (D/2) is always greater than
x, the expression inside the square root is always positive and thus
the solutions for x are always real. Eq. A3 can be rearranged by
factoring out x2 from inside the square root, using Pythagorean
trigonometric identities, and choosing the positive root so that the
y is positive:
y = x cos(π − ϕ + β ) +
D 2
2x
−
sin2
(π − ϕ + β )
(A4)
The horizon height of the rim is simply:
d
y
Hrim = arctan
(A5)
Combining Eqs. A4 and A5, this becomes:
⎡
⎤
d
Hrim = arctan⎣ ×
x
1
cos(π − ϕ + β ) +
D 2
2x
⎦
− sin2 (π − ϕ + β )
(A6)
Fig. A1. Diagram showing relevant variables for calculation of the horizon height of
the crater rim, Hrim . The red dot is the location of the observer on the crater floor,
and the red arrow points in the direction they are facing. See text S1 for explanation
of the variables. (For interpretation of the references to colour in this figure legend,
the reader is referred to the web version of this article.)
A.2. Horizon height: central peak
Assume an observer is standing on the floor of a complex crater
with central peak diameter Dpeak and central peak height hpeak at
a distance x from the center of the crater (Fig. A2). The observer is
located at an angle ϕ , measured clockwise from due north, and is
facing direction β measured the same way. The lateral distance y1
between the observer and the highest point of the central peak in
the direction the observer is facing is given by:
y1 = xcos(|π − β + ϕ | )
(A7)
Similarly, the lateral distance y2 between the center of the central peak and highest point of the central peak in the direction the
observer is facing is given by:
y2 = xsin(|π − β + ϕ | )
(A8)
The height of that point on the central peak, z, assuming the
peak is in the shape of a cone (i.e., has uniform slope) is:
z = h peak 1 −
2y2
D peak
(A9)
The horizon height of the rim is:
H peak = arctan
z
Combining Eqs. A7–A10, this becomes:
H peak = arctan
(A10)
y1
h peak
D peak − 2xsin(|π − β + ϕ | )
×
D peak
xcos(|π − β + ϕ | )
(A11)
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Fig. A2. Diagram showing relevant variables for calculation of the horizon height
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floor, and the red arrow points in the direction they are facing. See text S2 for
explanation of the variables. Angles ϕ and β are omitted for clarity, but represent
the same angles as shown in Figure S1.
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